page_title: Lie algebra extension
section_title: Current algebra
hierarchical_section_title: Lie algebra extension / Applications / Current algebra
caption_attribution_description: DAVOS/SWITZERLAND, 28JAN12 - Murray Gell-Mann, Distinguished Fellow, Santa Fe Institute, USA, is captured during the session 'The Secrets of the Universe' at the Annual Meeting 2012 of the World Economic Forum at the congress centre in Davos, Switzerland, January 28, 2012.. . Copyright by World Economic Forum. swiss-image.ch/Photo by Michael Wuertenberg
context_page_description: In the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extension e is an enlargement of a given Lie algebra g by another Lie algebra h. Extensions arise in several ways. There is the trivial extension obtained by taking a direct sum of two Lie algebras. Other types are the split extension and the central extension. Extensions may arise naturally, for instance, when forming a Lie algebra from projective group representations. Such a Lie algebra will contain central charges.
  Starting with a polynomial loop algebra over finite-dimensional simple Lie algebra and performing two extensions, a central extension and an extension by a derivation, one obtains a Lie algebra which is isomorphic with an untwisted affine Kac–Moody algebra. Using the centrally extended loop algebra one may construct a current algebra in two spacetime dimensions. The Virasoro algebra is the universal central extension of the Witt algebra.
context_section_description: As an application of a central extension of polynomial loop algebra, a current algebra of a quantum field theory is considered (background). Suppose one has a current algebra, with the interesting commutator being
  with a Schwinger term. To construct this algebra mathematically, let g be the centrally extended polynomial loop algebra of the previous section with
  as one of the commutation relations, or, with a switch of notation (l→m, m→n, i→a, j→b, λᵐ⊗Gₐ→Tᵐₐ) with a factor of i under the physics convention,
  Define using elements of g,
  One notes that
  so that it is defined on a circle. Now compute the commutator,
  For simplicity, switch coordinates so that y → 0, x → x − y ≡ z and use the commutation relations,
  Now employ the Poisson summation formula,
  for z in the interval (0, L) and differentiate it to yield
  and finally
  or
  since the delta functions arguments only ensure that the arguments of the left and right arguments of the commutator are equal (formally δ(z) = δ(z − 0) ↦  δ((x −y) − 0) = δ(x −y)).
  By comparison with CA10, this is a current algebra in two spacetime dimensions, including a Schwinger term, with the space dimension curled up into a circle. In the classical setting of quantum field theory, this is perhaps of little use, but with the advent of string theory where fields live on world sheets of strings, and spatial dimensions are curled up, there may be relevant applications.
caption: Murray Gell-Mann, 1969 Nobel Laureate in physics, initiated the field of current algebra in the 1960s. It exploits known local symmetries even without knowledge of the underlying dynamics to extract predictions, e.g. the Adler–Weisberger sum rule.
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